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**dwat** $\displaystyle C(t) = 38+2sin(\frac{\pi}{6}t)$ models a patient in degrees celsius during the 12 days of illness.

It asks to use the function C(t) to find the maximum temperature that the person reaches during the illness.

$\displaystyle C(t) = 38+2sin(\frac{\pi}{6}t)$

Differentiated and equate to zero

$\displaystyle C'(t)= 2cos\Big(\frac{\pi}{6}t\Big)*\frac{\pi}{6}$

$\displaystyle 0 = 2cos\Big(\frac{\pi}{6}t\Big)*\frac{\pi}{6}$

$\displaystyle t = \Big(\frac{cos^{-1}(0)}{\frac{\pi}{6}}\Big)$

$\displaystyle t = 3$

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$\displaystyle C(t) = 38+2sin(\frac{\pi}{6}3)$

$\displaystyle C(t) = 40$

My question is, I've found the maximum for the modelled function (3,40), now what if they asked to also find the minimum. I know the amplitude is 2 therefore the minimum will be 36 degrees celsius, (9, 36). However, that is just by observation of the features of the function. How do you find it algebraically?